Dimension2 $A_5$

Group $G=A_5$ has 2 irreducible 3-dimensional representations, one representation is as follows:

\[\begin{align*} G=\langle \begin{pmatrix} 1 & 0 & 0\\ 0 & \zeta_5^4 & 0\\ 0 & 0 & \zeta_5 \end{pmatrix}, \begin{pmatrix} -1 & 0 & 0\\ 0 & 0 & -1\\ 0 & -1 & 0 \end{pmatrix}, \frac{1}{\sqrt{5}} \begin{pmatrix} 1 & 1 & 1\\ 2 & s & t\\ 2 & t & s \end{pmatrix} \rangle. \end{align*}\]

where $\zeta_5$ is a 5-th root of unity, $s=\zeta_5^2+\zeta_5^3$ and $t=\zeta_5+\zeta_5^4$.

The Magma code is as follows:


FScale:=CyclotomicField(60);

F:=FScale;

Z5:=RootOfUnity(5);
s:=Z5^2+Z5^3;
t:=Z5+Z5^4;
sq5:=2*t+1;


G:=MatrixGroup<3,F|
[1,0,0, 0,Z5^4,0, 0,0,Z5],
[-1,0,0, 0,0,-1, 0,-1,0],
[1/sq5,1/sq5,1/sq5, 2/sq5,s/sq5,t/sq5, 2/sq5,t/sq5,s/sq5]>;

The Burnside symbol is

\[\begin{align*} [\mathbb{P}^2 \circlearrowleft G]=&(1,G \circlearrowright k(x,y),())\\ &+2(C_2,C_2 \circlearrowright k(x),(1))+(C_3,1 \circlearrowright k(x),(1,1))\\ &+(C_5,1 \circlearrowright k,(1,2))+(C_5,1 \circlearrowright k,(3,3))+(C_5,1 \circlearrowright k,(3,4))\\ &+(C_2^2,1 \circlearrowright k,((1,0),(1,1)))+(C_2^2,1 \circlearrowright k,((0,1),(1,0))) \end{align*}\]

For more detail, see file First $A_5$ Magma state

Another linear representation of $A_5$ is given by

\[\begin{align*} G=\langle \begin{pmatrix} 1 & 0 & 0\\ 0 & \zeta_5^3 & 0\\ 0 & 0 & \zeta_5^2 \end{pmatrix}, \begin{pmatrix} -1 & 0 & 0\\ 0 & 0 & -1\\ 0 & -1 & 0 \end{pmatrix}, \frac{1}{\sqrt{5}} \begin{pmatrix} -1 & -1 & -1\\ -2 & -t & -s\\ -2 & -s & -t \end{pmatrix} \rangle. \end{align*}\]

where $\zeta_5$ is a 5-th root of unity, $s=\zeta_5^2+\zeta_5^3$ and $t=\zeta_5+\zeta_5^4$.

The Magma code is as follows:


FScale:=CyclotomicField(60);

F:=FScale;

Z5:=RootOfUnity(5);
s:=Z5^2+Z5^3;
t:=Z5+Z5^4;
sq5:=2*t+1;


G:=MatrixGroup<3,F|
[1,0,0, 0,Z5^3,0, 0,0,Z5^2],
[-1,0,0, 0,0,-1, 0,-1,0],
[-1/sq5,-1/sq5,-1/sq5, -2/sq5,-t/sq5,-s/sq5, -2/sq5,-s/sq5,-t/sq5]>;

The Burnside symbol is

\[\begin{align*} [\mathbb{P}^2 \circlearrowleft G]=&(1,G \circlearrowright k(x,y),())\\ &+2(C_2,C_2 \circlearrowright k(x),(1))+(C_3,1 \circlearrowright k(x),(2,2))\\ &+2(C_5,1 \circlearrowright k,(1,2))+(C_5,1 \circlearrowright k,(3,3))\\ &+(C_2^2,1 \circlearrowright k,((1,0),(1,1)))+(C_2^2,1 \circlearrowright k,((0,1),(1,0))) \end{align*}\]

For more detail, see file Second $A_5$ Magma state

These 2 Burnside classes are the same.

Kaiqi Yang(杨凯奇)